Curvature tensor and intrinsic geometry of surfaces: Equations of Gauss and Weingarten, Theorema egregium, Fundamental theorem of the local theory of surfaces (Sections 4A-D, Kuehnel)
More on the curvature tensor and Weingarten map, Sectional curvature, Ricci tensor and curvature, Einstein tensor (Chapters 4,6: Kuehnel)
Spaces of constant curvature: Hyperbolic space, Geodesics and Jacobi fields, Local isometry of spaces of constant curvature (Chapter 7: Kuehnel)
Defense training
WS 2018/2019
Analysis and Probability
Organizational meeting
Probability that random polynomial has no real roots (Talk by Anna Gusakova)
Theorems on positive real functions (Talk by Anna Muranova)
Non-explosion of solutions to SDEs with singular coefficients (Talk by Chengcheng Ling)
Emergence of Flocking in the Fractional Cucker-Smale Model (Talk by Peter Kuchling)
Robust Poincaré inequality and Application to the transitional phase of fractional Laplacian (Talk by Guy Fabrice Foghem Gounoue)
My problems and failed ideas regarding the edge fluctuation of non hermitian random matrices with independent entries (Jonas Jalowy)
Effective impedance of a finite electric network. Definitions and main properties (theorems and conjectures) (Anna Muranova)
Application of Probability Methods in Number Theory and Integral Geometry (Anna Gusakova)
Ordered fields. The ordered field of rational functions“ (Anna Muranova)
Dynamics on the cone: an overview of my thesis (Peter Kuchling)
Well-posedness of SDE driven by Levy noise (Chengcheng Ling)
Discussion about the talks for workshop/retreat
Statistical Mechanics
Point processes 3: Moment Problem, Local Convergence Jansen, Gibbsian Point Processes, pp. 33 - 45
Point processes 2: Intensity Measure, Correlation Functions, Generating Functionals Jansen, Gibbsian Point Processes, pp. 25 - 33
Point processes 1: Configuration Spaces, Observables, PPP Jansen, Gibbsian Point Processes, pp. 15 - 25
Introduction to partition functions
Partition Functions and Gibbs Measures
Correlation Functions
Phase Transitions (Critical Exponents and Models)
Scaling Limits
Mean Field Limit of the Cucker-Smale model: Two Approaches (Peter Kuchling)
WS 2017/2018
Graph Theory
ReadingSpectral graph theory by Fan R. K. Chung
Chapter 2: Isoperimetric problems
Chapter 3: Diameters and eigenvalues
Talk by Anna Muranova: On effective resistance of electric network with impedances
Talk by Melissa Meinert: Ollivier Ricci curvature for general Laplacians
Markov chains
Talk by Filip Bosnic: Examples of Poincaré and log-Sobolev inequalities on discrete configuration spaces
Introduction to random graphs by Alan Frieze and Michał Karoński
Lévy processes
DiscussLévy Processes and Infinitely Divisible Distributions by Ken-iti Sato
Chapter 1 (pp. 1 - 30): Examples of Levy Processes
Chapter 2 (pp. 31 - 68): Characterisation and Existence of Levy Processes and Additive Processes
Chapter 2 (pp. 31 - 41): Infinitely Divisible Processes and the Lévy-Khintchine formula
Chapter 2 (pp. 54 - 67): Transition Functions and the Markov Property; Existence of Lévy and Additive Processes
Chapter 3 (pp. 69 - 77): Selfsimilar and Semi-selfsimilar Processes and their Exponents
Chapter 3 (pp. 77 - 90): Representations of Stable and Semi-stable Distributions
Viswanathan et al. Papers: “Levy Flight Search Patterns of Wandering Albatrosses” and “Optimizing the Success of Random Searches”
Bertoin (pp. 18-24): Potential theory: Markov property and important definitions
